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Mirrors > Home > MPE Home > Th. List > 0dgr | Structured version Visualization version GIF version |
Description: A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
0dgr | ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3587 | . . . 4 ⊢ ℂ ⊆ ℂ | |
2 | plyconst 23766 | . . . 4 ⊢ ((ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ) → (ℂ × {𝐴}) ∈ (Poly‘ℂ)) | |
3 | 1, 2 | mpan 702 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) ∈ (Poly‘ℂ)) |
4 | 0nn0 11184 | . . . 4 ⊢ 0 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℕ0) |
6 | simpl 472 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ (0...0)) → 𝐴 ∈ ℂ) | |
7 | 0z 11265 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
8 | exp0 12726 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (𝑧↑0) = 1) | |
9 | 8 | oveq2d 6565 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (𝐴 · (𝑧↑0)) = (𝐴 · 1)) |
10 | mulid1 9916 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) | |
11 | 9, 10 | sylan9eqr 2666 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝐴 · (𝑧↑0)) = 𝐴) |
12 | simpl 472 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝐴 ∈ ℂ) | |
13 | 11, 12 | eqeltrd 2688 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝐴 · (𝑧↑0)) ∈ ℂ) |
14 | oveq2 6557 | . . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑧↑𝑘) = (𝑧↑0)) | |
15 | 14 | oveq2d 6565 | . . . . . . . 8 ⊢ (𝑘 = 0 → (𝐴 · (𝑧↑𝑘)) = (𝐴 · (𝑧↑0))) |
16 | 15 | fsum1 14320 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ (𝐴 · (𝑧↑0)) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)) = (𝐴 · (𝑧↑0))) |
17 | 7, 13, 16 | sylancr 694 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)) = (𝐴 · (𝑧↑0))) |
18 | 17, 11 | eqtrd 2644 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)) = 𝐴) |
19 | 18 | mpteq2dva 4672 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ 𝐴)) |
20 | fconstmpt 5085 | . . . 4 ⊢ (ℂ × {𝐴}) = (𝑧 ∈ ℂ ↦ 𝐴) | |
21 | 19, 20 | syl6reqr 2663 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(𝐴 · (𝑧↑𝑘)))) |
22 | 3, 5, 6, 21 | dgrle 23803 | . 2 ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) ≤ 0) |
23 | dgrcl 23793 | . . 3 ⊢ ((ℂ × {𝐴}) ∈ (Poly‘ℂ) → (deg‘(ℂ × {𝐴})) ∈ ℕ0) | |
24 | nn0le0eq0 11198 | . . 3 ⊢ ((deg‘(ℂ × {𝐴})) ∈ ℕ0 → ((deg‘(ℂ × {𝐴})) ≤ 0 ↔ (deg‘(ℂ × {𝐴})) = 0)) | |
25 | 3, 23, 24 | 3syl 18 | . 2 ⊢ (𝐴 ∈ ℂ → ((deg‘(ℂ × {𝐴})) ≤ 0 ↔ (deg‘(ℂ × {𝐴})) = 0)) |
26 | 22, 25 | mpbid 221 | 1 ⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 ≤ cle 9954 ℕ0cn0 11169 ℤcz 11254 ...cfz 12197 ↑cexp 12722 Σcsu 14264 Polycply 23744 degcdgr 23747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-0p 23243 df-ply 23748 df-coe 23750 df-dgr 23751 |
This theorem is referenced by: 0dgrb 23806 coemulc 23815 dgr0 23822 dgrmulc 23831 dgrcolem2 23834 plyremlem 23863 vieta1lem2 23870 |
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